Quantitative bounds for large deviations of heavy tailed random variables

TL;DR

This paper provides precise quantitative bounds for the probability of large deviations in sums of heavy-tailed random variables and characterizes the distribution of summands conditioned on large sums.

Contribution

It introduces explicit bounds for large deviation probabilities and describes the conditioned law of summands, advancing understanding of heavy-tailed sum behaviors.

Findings

Large deviation probability approximated by single summand event

Quantitative bounds on approximation error

Characterization of summand distribution conditioned on large sum

Abstract

The probability that the sum of independent, centered, identically distributed, heavy-tailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We quantify the error in this approximation. We furthermore characterise of the law of the individual summands, conditioned on the sum being large.